$(function() {

  // initialize
  $('.materialSelect').material_select();

  // setup listener for custom event to re-initialize on change
  $('.materialSelect').on('contentChanged', function() {
    $(this).material_select();
  });

  // update function for demo purposes
  $("#myButton").click(function() {
    
     // add new value
    var newValue = getNewDoggo();
    var $newOpt = $("<option>").attr("value",newValue).text(newValue)
    $("#myDropdown").append($newOpt);

    // fire custom event anytime you've updated select
    $("#myDropdown").trigger('contentChanged');
    
  });

});

function getNewDoggo() {
  var adjs =  ['Floofy','Big','Cute','Cuddly','Lazy'];
  var nouns = ['Doggo','Floofer','Pupper','Fluffer', 'Nugget'];
  var newOptValue = adjs[Math.floor(Math.random() * adjs.length)] + " " + 
                    nouns[Math.floor(Math.random() * nouns.length)];
  return newOptValue;
}

body { padding: 25px}

<link  href="https://cdnjs.cloudflare.com/ajax/libs/materialize/0.96.1/css/materialize.min.css" rel="stylesheet">
<script src="//cdnjs.cloudflare.com/ajax/libs/jquery/2.1.3/jquery.min.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/materialize/0.96.1/js/materialize.min.js"></script>

<button class="waves-effect waves-light btn" id="myButton">
  Add New Option to Dropdown
</button>

<select id="myDropdown" class="materialSelect">
  <option value="Happy Floof">Happy Floof</option>
  <option value="Derpy Biscuit">Derpy Biscuit</option>
</select>